While the axioms of cohesion do imply the intrinsic existence of exponentiatedinfinitesimal spaces, they do not admit access to an explicit synthetic notion of infinitesimal extension.

Here we consider one extra axiom on a cohesive (∞,1)-topos that does imply a good intrinsic notion of synthetic differential extension, compatible with the given notion of cohesion. We speak of differential cohesion.

In a cohesive (∞,1)(\infty,1)-topos with differential cohesion there are for instance good intrinsic notions of formal smoothness and of de Rham spaces of objects.

Differential cohesion

We discuss extra structure on a cohesive (∞,1)-topos that encodes a refinement of the corresponding notion of cohesion to infinitesimal cohesion . More precisely, we consider inclusions H↪Hth\mathbf{H} \hookrightarrow \mathbf{H}_{th} of cohesive

(∞,1)(\infty,1)-toposes that exhibit the objects of Hth\mathbf{H}_{th} as infinitesimal cohesive neighbourhoods of objects in H\mathbf{H}.

Definition

Definition

Given a cohesive (∞,1)(\infty,1)-topos H\mathbf{H} we say that an infinitesimal cohesive neighbourhood of H\mathbf{H} is another cohesive (∞,1)(\infty,1)-topos Hth\mathbf{H}_{th} equipped with an adjoint quadruple of adjoint (∞,1)-functors of the form

Remark

Proof

By the characterizaton of full and faithful adjoint (∞,1)-functors the condition on i!i_! is equivalent to i*i!≃Idi^* i_! \simeq Id. Since (i*i!⊣i*i*)(i^* i_! \dashv i^* i_*) it follows by essential uniqueness of adjoint (∞,1)-functors that also i*i*≃Idi^* i_* \simeq Id.

Remark

This definition captures the characterization of an infinitesimal object as having a single global point surrounded by an infinitesimal neighbourhood: as we shall see in more detail below, the (∞,1)-functori*i^* may be thought of as contracting away any infinitesimal extension of an object. Thus XX being an infinitesimal object amounts to i*X≃*i^* X \simeq *, and the (∞,1)-adjunction(i!⊣i*)(i_! \dashv i^*) then indeed guarantees that XX has only a single global point, since

Consider the right Kan extensionRani:[Cop,sSet]→[Cthop,sSet]Ran_i : [C^{op}, sSet] \to [C_{th}^{op},sSet] of simplicial presheaves along the functor ii. On an object K×D∈CthK \times D \in C_{th} it is given by the end-expression

and by Yoneda reduction (more explicitly: observing that this is equivalently the formula for left Kan extension of the non-corepresentable Cth(K×D,i(−)):C→sSetC_{th}(K \times D, i(-)) : C \to sSet along the identity functor) this is

We first check that (−)∘i(-) \circ i sends locally fibrant objects to locally fibrant objects.

To that end, let {Ui→U}\{U_i \to U\} be a covering family in CC. Write ∫[k]∈ΔΔ[k]⋅∐i0,⋯,ik(j(Ui0)×j(U)j(Ui1)×j(U)⋯×j(U)j(Uk))\int^{[k] \in \Delta} \Delta[k] \cdot \coprod_{i_0, \cdots, i_k} (j(U_{i_0}) \times_{j(U)} j(U_{i_1}) \times_{j(U)} \cdots \times_{j(U)} j(U_k)) for its Cech nerve, where jj denotes the Yoneda embedding. Recall by the definition of the ∞-cohesive siteCC that all the fiber products of representable presheaves here are again themselves representable, hence ⋯=∫[k]∈ΔΔ[k]⋅∐i0,⋯,ik(j(Ui0×UUi1×U⋯×UUk))\cdots = \int^{[k] \in \Delta} \Delta[k] \cdot \coprod_{i_0, \cdots, i_k} (j(U_{i_0} \times_U U_{i_1} \times_U \cdots \times_U U_k)). This means that the left adjointLaniLan_i preserves not only the coend and tensoring, but by the remark in the previous paragraph and the assumption that ii preserves pullbacks along covers we have that

By the assumption that ii preserves covers, this is the Cech nerve of a covering family in CthC_{th}. Therefore for F∈[Cthop,sSet]proj,locF \in [C_{th}^{op}, sSet]_{proj,loc} fibrant we have for all coverings {Ui→U}\{U_i \to U\} in CC that the descent morphism

To see that (−)∘p(-) \circ p preserves locally fibrant objects, we apply the analogous reasoning after observing that its left adjoint(−)∘i(-)\circ i preserves all limits and colimits of simplicial presheaves (as these are computed objectwise) and by observing that for {Ui→piU}\{\mathbf{U}_i \stackrel{p_i}{\to} \mathbf{U}\} a covering family in CthC_{th} we have that its image under (−)∘i(-) \circ i is its image under pp, by the Yoneda lemma:

Therefore (−)∘i(-) \circ i is a left and right local Quillen functor with left local Quillen adjoint LaniLan_i and right local Quillen adjoint (−)∘p(-)\circ p.

It follows that i*:Sh(∞,1)(Cth)→Sh(∞,1)(C)i^* : Sh_{(\infty,1)}(C_{th}) \to Sh_{(\infty,1)}(C) is given by the left derived functor of restriction along ii, and i*:Sh(∞,1)(C)→Sh(∞,1)(Cth)i_* : Sh_{(\infty,1)}(C) \to Sh_{(\infty,1)}(C_{th}) is given by the right derived functor of restriction along pp.

Finally to see that also RanpRan_p preserves locally fibrant objects by the same reasoning as above, notice that for every covering family {Ui→U}\{U_i \to U\} in CC and every morphism K→p*U\mathbf{K} \to p^* U in CthC_{th} we may find a covering {Kj→K}\{\mathbf{K}_j \to \mathbf{K}\} of K\mathbf{K} such that we find commuting diagrams on the left of

It remains to see that i!i_! is full and faithful. For that notice the general fact that left Kan extension (see the properties discussed there) along a full and faithful functorii satisfies Lani∘i≃idLan_i \circ i \simeq id. It remains to observe that since (−)∘i(-)\circ i is not only right but also left Quillen by the above, we have that i*Lanii^* Lan_i applied to a cofibrant object is already the derived functor of the composite.

Remark

In traditional contexts see (SimpsonTeleman, p. 7) the object ℑ(X)\Im(X) is called the de Rham space of XX or the de Rham stack of XX . Here we may tend to avoid this terminology, since by the discussion at cohesive (∞,1)-topos – de Rham cohomology we have a good notion of intrinsic de Rham cohomology in any cohesive (∞,1)-topos already without equipping it with differential cohesion. From this point of view the object ℑ(X)\Im(X) is not primarily characterized by the fact that (in some models, see below) it does co-represent de Rham cohomology – because the object ΠdR(X)\mathbf{\Pi}_{dR}(X) from above does, too – but by the fact that it does so in an explicitly (synthetic) infinitesimal way.

Proof

This is the formula for the unit of the composite adjunction Hth←Discinf→ΠinfH←Disc→Π∞Grpd\mathbf{H}_{th} \stackrel{\overset{\Pi_{inf}}{\to}}{\underset{Disc_{inf}}{\leftarrow}} \mathbf{H} \stackrel{\overset{\Pi}{\to}}{\underset{Disc}{\leftarrow}} \infty Grpd:

Proof

By the condition that i!i_! is a full and faithful (∞,1)-functor the second morphism here in an equivalence, as indicated, and hence the component of the composite on XX being an effective epimorphism is equivalent to the component i!X→Πi!Xi_! X \to \mathbf{\Pi} i_! X being an effective epimorphism.

The (∞,1)-pullback of a formally étale morphisms is formally étale if the pullback is preserved by i!i_!.

The statements about closure under composition and pullback appears as(KontsevichRosenberg, prop. 5.4, prop. 5.6). Notice that the extra assumption that i!i_! preserves the pullback is implicit in their setup, by remark 8.

Proof

The first statement follows since ∞\infty-pullbacks are well defined up to quivalence.

If ff and gg are formally étale then both small squares are pullback squares. Then the pasting law says that so is the outer rectangle and hence g∘fg \circ f is formally étale. Similarly, if gg and g∘fg \circ f are formally étale then the right square and the total reactangle are pullbacks, so the pasting law says that also the left square is a pullback and so also ff is formally étale.

For the fourth claim, let Id≃(g→f→g)Id \simeq (g \to f \to g) be a retract in the arrow (∞,1)-categoryHI\mathbf{H}^I. By applying the natural transformation ϕ:i!→I*\phi : i_! \to I_* we obtain a retract

in the category of squares H□\mathbf{H}^{\Box}. We claim that generally, if the middle piece in a retract in H□\mathbf{H}^\Box is an (∞,1)-pullback square, then so is its retract sqare. This implies the fourth claim.

One way to motivate this is to consider structure sheaves of flat differential forms. To that end, let G∈Grp(Hth)G \in Grp(\mathbf{H}_{th}) a differential cohesive ∞-group with de Rham coefficient object♭dRBG\flat_{dR}\mathbf{B}G and for X∈HthX \in \mathbf{H}_{th} any differential homotopy type, the product projection

X×♭dRBG→X
X \times \flat_{dR} \mathbf{B}G \to X

regarded as an object of the slice (∞,1)-topos(Hth)/X(\mathbf{H}_{th})_{/X}almost qualifies as a “bundle of flat 𝔤\mathfrak{g}-valued differential forms” over XX: for U→XU \to X an cover (a 1-epimorphism) regarded in (Hth)/X(\mathbf{H}_{th})_{/X}, a UU-plot of this product projection is a UU-plot of XX together with a flat 𝔤\mathfrak{g}-valued de Rham cocycle on XX.

This is indeed what the sections of a corresponding bundle of differential forms over XX are supposed to look like – but only ifU→XU \to X is sufficiently spread out over XX, hence sufficiently étale. Because, on the extreme, if XX is the point (the terminal object), then there should be no non-trivial section of differential forms relative to UU over XX, but the above product projection instead reproduces all the sections of ♭dRBG\flat_{dR} \mathbf{B}G.

In order to obtain the correct cotangent-like bundle from the product with the de Rham coefficient object, it needs to be restricted to plots out of suficiently étale maps into XX. In order to correctly test differential form data, “suitable” here should be “formally”, namely infinitesimally. Hence the restriction should be along the full inclusion

of the formally étale maps of def. 8 into XX. Since on formally étale covers the sections should be those given by ♭dRBG\flat_{dR}\mathbf{B}G, one finds that the corresponding “cotangent bundle” must be the coreflection along this inclusion. The following proposition establishes that this coreflection indeed exists.

Proof

By the general discussion at reflective factorization system, the reflection is given by sending a morphism f:Y→Xf \colon Y \to X to X×ℑ(X)ℑ(Y)→YX \times_{\Im(X)} \Im(Y) \to Y and the reflection unit is the left horizontal morphism in

So consider any diagram(∞,1)-functorI→(Hth)/XfetI \to (\mathbf{H}_{th})_{/X}^{fet} out of a small (∞,1)-category. Since the inclusion of (Hth)/Xfet(\mathbf{H}_{th})_{/X}^{fet} is full, it is sufficient to show that the (∞,1)(\infty,1)-colimit over this diagram taken in (Hth)/X(\mathbf{H}_{th})_{/X} lands again in (Hth)/Xfet(\mathbf{H}_{th})_{/X}^{fet} in order to have that (∞,1)(\infty,1)-colimits are preserved by the inclusion. Moreover, colimits in a slice of Hth\mathbf{H}_{th} are computed in Hth\mathbf{H}_{th} itself (this is discussed at slice category - Colimits).

This diagram is now indeed an (∞,1)-pullback by the fact that we have universal colimits in the (∞,1)-toposHth\mathbf{H}_{th}, hence that on the left the component YiY_i for each i∈Ii \in I is the (∞,1)-pullback of ℑ(Yi)→ℑ(X)\Im(Y_i) \to \Im(X), by assumption that we are taking an (∞,1)(\infty,1)-colimit over formally étale morphisms.

Example

For the case that X≃*X \simeq \ast in prop. 8, then the proof there shows that the étalification operation over the point is just &{\&} :

Remark

For U∈HthU \in \mathbf{H}_{th} a test object (say an object in a (∞,1)-site of definition, under the Yoneda embedding) a formally étale morphism U→XU \to X is like an open map/open embedding. Regarded as an object in (Hth)/Xfet(\mathbf{H}_{th})_{/X}^{fet} we may consider the sections over UU of the cotangent bundle as defined above, which in Hth\mathbf{H}_{th} are diagrams

where we are now simply including on the left the formally étale map (U→X)(U \to X) along (Hth)/Xfet↪(Hth)/X(\mathbf{H}_{th})^{fet}_{/X} \hookrightarrow (\mathbf{H}_{th})_{/X}.

In other words, the sections of the GG-valued flat cotangent sheaf 𝒪X(♭dRBG)\mathcal{O}_X(\flat_{dR}\mathbf{B}G) are just the sections of X×♭dRBG→XX \times \flat_{dR}\mathbf{B}G \to X itself, only that the domain of the section is constrained to be a formally étale patch of XX.

But then by the very nature of ♭dRBG\flat_{dR}\mathbf{B}G it follows that the flat sections of the GG-valued cotangent bundle of XX are indeed nothing but the flat GG-valued differential forms on XX.

Proposition

For X∈HthX \in \mathbf{H}_{th} an object in a differentially cohesive ∞\infty-topos, then its petit structured ∞\infty-topos ShHth(X)Sh_{\mathbf{H}_{th}}(X), according to def. 11, is locally ∞-connected.

preserves (∞,1)-limits, so that it has a further left adjoint. Here LL is the reflector from prop. 8. Inspection shows that this composite sends an object A∈∞GrpdA \in \infty Grpd to ℑ(Disc(A))×X→X\Im(Disc(A)) \times X \to X:

Now for A:J→∞GrpdA \colon J \to \infty Grpd a diagram, it is taken to the diagram j↦ℑ(Disc(Aj))×X→Xj \mapsto \Im(Disc(A_j)) \times X \to X in ShH(X)Sh_{\mathbf{H}}(X) and so its ∞\infty-limit is computed in H\mathbf{H} over the diagram locally of the form

For the equivalence of structure sheaves it is sufficient to show for each coefficientA∈HthA \in \mathbf{H}_{th} an equivalence

𝒪Y(A)≃(f*𝒪X(A))
\mathcal{O}_Y(A) \simeq (f^\ast \mathcal{O}_X(A))

in ShH(Y)Sh_{\mathbf{H}}(Y). But by definition (11) 𝒪Y(A)≔Et(A×Y)\mathcal{O}_Y(A) \coloneqq Et(A \times Y) and similarly for 𝒪X\mathcal{O}_X and since EtEt is right adjoint to the inclusion ShH(Y)↪HYSh_{\mathbf{H}}(Y) \hookrightarrow \mathbf{H}_{Y} we have

for the morphism in H\mathbf{H} which is the (∑X⊣X*)(\underset{X}{\sum} \dashv X^*)-adjunct ∑XιEt(X*A)→A\underset{X}{\sum}\iota Et(X^* A) \to A of the counitιEt(X*A)→X*A\iota Et(X^* A) \to X^* A of the (ι⊣Et)(\iota \dashv Et)-coreflection of def. 11.

This θX(A)\theta_X(A) we call the Liouville-Poincaré AA-cocycle on ∑Xι𝒪X(A)\underset{X}{\sum} \iota \mathcal{O}_X(A).

Example

Consider the model of differential cohesion given by Hth=\mathbf{H}_{th} =SynthDiff∞Grpd. Write Ω1∈H↪i!Hth\Omega^1 \in \mathbf{H }\stackrel{i_!}{\hookrightarrow} \mathbf{H}_{th} for the abstract sheaf of differential 1-forms.

Then for X∈SmthMfd↪HX \in SmthMfd \hookrightarrow \mathbf{H} a smooth manifold, we have that

of the manifold: because for iU:U→Xi_U \colon U \to X an open subset of the manifold regarded as an object of ShH(X)Sh_{\mathbf{H}}(X), a section ι(σU)\iota(\sigma_U) of T*X|U→UT^* X|_U \to U is equivalently a map σ:iU→𝒪X(Ω1)\sigma \colon i_U \to \mathcal{O}_X(\Omega^1) in ShHth(X)Sh_{\mathbf{H}_{th}}(X), which by the (ι⊣Et)(\iota \dashv Et)-adjunction is a map ι(iU)→X×Ω1\iota(i_U) \to X \times \Omega^1 in (Hth)/X(\mathbf{H}_{th})_{/X} which finally is equivalently a map U→Ω1U \to \Omega^1 in Hth\mathbf{H}_{th} hence an element in Ω1(U)\Omega^1(U).

hence the original σ\sigma. This is the defining property which identifies that\that as the traditional Liouville-Poincaré 1-form.

Manifolds and étale groupoids

An ordinary topological/Lieétale groupoid is one whose source/target map is an étale map. We consider now a notion that can be formulated in the presence of infinitesimal cohesion which generalizes this.

Definition

Remark

When the infinitesimal shape modality exhibits first-order infinitesimals, such that 𝔻(V)\mathbb{D}(V) is the first order infinitesimal neighbourhood of a point, then Aut(𝔻(V))\mathbf{Aut}(\mathbb{D}(V)) indeed plays the role of the general linear group. When 𝔻n\mathbb{D}^n is instead a higher order or even the whole formal neighbourhood, then GL(n)GL(n) is rather a jet group. For order kk-jets this is sometimes written GLk(V)GL^k(V) We nevertheless stick with the notation “GL(V)GL(V)” here, consistent with the fact that we have no index on the infinitesimal shape modality. More generally one may wish to keep track of a whole tower of infinitesimal shape modalities and their induced towers of concepts discussed here.

This class of examples of framings is important:

Proposition

Every differentially cohesive ∞-groupGG is canonically framed (def. 17) such that the horizontal map in def. 16 is given by the left action of GG on its infinitesimal disk at the neutral element:

where the right square is the defining pullback for the infinitesimal disk𝔻G\mathbb{D}^G. For the left square we find by this proposition that TinfG≃G×𝔻GT_{inf} G \simeq G\times \mathbb{D}^G and that the top horizontal morphism is as claimed.

Proposition

For VV a framed object, def. 17, let XX be a VV-manifold, def. 15. Then the infinitesimal disk bundle, def. 16, of XX canonically trivializes over any VV-cover V←U→XV \leftarrow U \rightarrow X , i.e. there is a homotopy pullback of the form

(a diagram in H/BGL(V)\mathbf{H}_{/\mathbf{B}GL(V)}) extends to a sliced correspondence between c\mathbf{c} and the trivial GG-structure c0\mathbf{c}_0 on VV, example 5, hence to a diagram in H/BGL(V)\mathbf{H}_{/\mathbf{B}GL(V)} of the form

On the other hand, c\mathbf{c} is called infinitesimally integrable (or torsion-free) if such an extension exists (only) after restriction to all infinitesimal disks in XX and UU, hence after composition with the counit

♭relU⟶U
\flat^{rel} U \longrightarrow U

of the relative flat modality, def. 3 (using that by prop. 4 this is also formally étale and hence induces map of frame bundles):

Note

The objects on the left are principal ∞-bundles equipped with flat ∞-connection . The first morphism forgets their higher parallel transport along finite volumes and just remembers the parallel transport along infinitesimal volumes. The last morphism finally forgets also this connection information.

where on the right we have ordinary cohomology in Top (for instance realized as singular cohomology) with coefficients in the discrete groupAdisc:=ΓAA_{disc} := \Gamma A underlying the cohesive group AA.

In certain contexts of infinitesimal neighbourhoods of cohesive ∞\infty-toposes the de Rham theorem in this form has been considered in (SimpsonTeleman).

It follows that every such ∞\infty-Lie algebroid X→𝒢X \to \mathcal{G} canonically maps to the tangent ∞\infty-Lie algebroid of XX – the anchor map. The naturality square of the unit ηpℑ\eta^{\Im}_{p} exhibits the morphism:

Lie theory

(…)

The discussion at synthetic differential ∞-groupoid – Lie differentiation immediately generalizes to produce a concept of Lie differentiation in any differentially cohesive context. This Lie differentiation is just the flat modality of the differential cohesion but regarded as cohesive over its induced infinitesimal cohesion. As such, there is a left adjoint to Lie differentiation, given by the corresponding shape modality. However, the substance of Lie theory here will be to restrict this adjunction to geometric ∞-stacks. On the geometric ∞\infty-stacks the Lie differentiation via passage to inffinitesimal cohesion will yield actual L∞L_\infty-algebras, but some structure is required to make the formal Lie integration of these lang indeed in geometric ∞-stacks.

The image of ii is contained in that of Ω∞\Omega^\infty. Therefore we may restrict the (cod⊣i)(cod \dashv i)-adjunction on the right to the full sub-(∞,1)-categoryT˜C\tilde T_C of CΔ[1]C^{\Delta[1]} on thise objects in the image of Ω∞\Omega^\infty. This yields an infinitesimal neighbourhood of (∞,1)-sites